Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-09T08:12:14.942Z Has data issue: false hasContentIssue false

Absolutely divergent series and classes of Banach spaces

Published online by Cambridge University Press:  24 October 2008

William H. Ruckle
Affiliation:
Department of Mathematical Sciences, Clemson University, Clemson, SC 29631, U.S.A.

Extract

A normed space E is said to be series immersed in a Banach space X if for every absolutely divergent series nxn in E there is a continuous linear mapping T from E into X such that nTxn diverges absolutely. The theorem of Dvoretzky and Rogers(1) implies that a normed space E is series immersed in a finite dimensional space if and only if E itself is finite dimensional. In (4) and (9) it was shown that E is series immersed in lp 1 p < if and only if E is isomorphic to a subspace of Lp() for some measure . In particular, E is isomorphic to an inner product space if and only if it is series immersed in a Hilbert space. The property of series immersion was further studied in the papers (7) and (8). The main results in these two papers are conditions on X under which E series immersed in X would imply E locally immersed in X, a condition slightly stronger (formally) than E finitely representable in X.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1983

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Dvoretzky, A. and Rogees, C. A.Absolute and unconditional convergence in normed linear spaces. Proc. Nat. Acad. Sci. U.S.A. 36 (1950), 192197.Google Scholar
(2)Grothendieck, A.Sur certaines classes de suites dans les espaces de Banach, et le thorme de DvoretzkyRogers. Bolm. Soc. Mat. S. Paulo 8 (1956), 81110.Google Scholar
(3)Jamison, R. E. and Ruckle, W. H.Factoring absolutely convergent series. Math. Ann. 224 (1976), 143148.CrossRefGoogle Scholar
(4)Kalton, N. J. and Ruckle, W. H.A series characterization of subspaces of Lp()-spaces. Bull. Amer. Math. Soc. 79 (1973), 10191022.CrossRefGoogle Scholar
(5)Pietsch, A.Absolut p-summierende Abbildugen in normierten Rumen. Studia Math. 28 (1967), 333353.Google Scholar
(6)Operator, Pietsch A.ideals (North-Holland, 1980).Google Scholar
(7)Ruckle, W. H.Absolutely divergent series and isomorphism of subspaces. Pacific J. Math. 58 (1975), 605615.Google Scholar
(8)Ruckle, W. H.Absolutely divergent series and isomorphism of subspaces II. Pacific J. Math. 68 (1977), 229240.Google Scholar
(9)Saphar, P.Une charactrisation des sousespaces de L p et ses applications. C. B. Acad. Sci. Paris 227 (1973), 3539.Google Scholar